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Homotopy and Isotopy Properties of Topological Spaces HOMOTOPY AND ISOTOPY PROPERTIES OF TOPOLOGICAL SPACES SZE-TSEN HU 1. Introduction. The most important notion in topology is that of a homeomorphism f: X —> F from a topological space X onto a topological space F. If a homeomorphism/: X —» F exists, then the topological spaces X and F are said to be homeomorphic (or topologically equivalent), in symbols, X = Y. The relation = among topological spaces is obviously reflexive, symmetric, and transitive; hence it is an equivalence relation. For an arbitrary family F of topological spaces, this equivalence relation = divides /Mnto disjoint equiva­ lence classes called the topology types of the family F. Then, the main problem in topology is the topological classification problem formulated as follows. The topological classification problem: Given a family F oi topological spaces, find an effective enumeration of the topology types of the family F and exhibit a representative space in each of these topology types. A number of special cases of this problem were solved long ago. For example, the family of Euclidean spaces is classified by their dimensions and the family of closed surfaces is classified by means of orientability and Euler characteristic. However, the problem is far from being solved; in fact, the topological classi­ fication of the family of three-dimensional compact manifolds still remains an outstanding unsolved problem. To overcome the difficulty of the topological classification problem, topo- logists introduced weaker equivalence relations, namely, the homotopy and isotopy equivalences, which would give rise to larger but fewer classes of spaces than the topology types. A continuous map /: X —> F is said to be a homotopy equivalence provided that there exists a continuous map g: Y —» X such that the compositions gof and fog are homotopic to the identity maps on X and F respectively. Two topological spaces X and F are said to be homotopically equivalent (in symbol, X ~ F) if there exists a homotopy equivalence /: X —> F. It is easily verified that the relation c^ among topological spaces is reflexive, symmetric, and transitive; hence it is an equivalence relation. For any given family F of topological spaces, this equivalence relation ~ divides F into dis­ joint equivalence classes called the homotopy types of the family. Analogous to Received November 16, 1959. This research was supported by the United States Air Force through the Air Force Office of Scientific Research and Development Command, under Contract No. AF 49(638)-179. Reproduction in whole or in part is permitted for any purpose of the United States Government. 167 Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 00:23:12, subject to the Cambridge Core terms of use. 168 SZE-TSEN HU the topological classification problem, one can formulate the homotopy classi­ fication problem in the obvious fashion. To introduce the notion of isotopy equivalence, let us first recall the defini­ tion of an imbedding. A continuous map /: X —> F is said to be an imbedding provided that / is a homeomorphism of X onto the subspace f(X) of F. A homotopy ht: X —> F, (t 6 /), is said to be an isotopy if, for each t Ç /, ht is an imbedding. Two imbeddings/, g: X —> Fare said to be isotopic if there exists an isotopy ht: X —• F, (/ Ç /), such that h0 = f and &i = g. An imbedding/: X —» F is said to be an isotopy equivalence if there exists an imbedding g: F—> X such that the composite imbeddings go/ and /o g are isotopic to the identity imbeddings on X and F respectively. Two topological spaces X and F are said to be isotopically equivalent (in symbol, X == F) if there exists an isotopy equivalence f:X—> Y. The relation = among topological spaces is obviously an equivalence rela­ tion. For any given family F of topological spaces, this equivalence relation = divides F into disjoint equivalence classes called the isotopy types of the family. One can formulate the isotopy classification problem in the obvious fashion. By the definitions given above, it is clear that every homeomorphism is an isotopy equivalence and that every isotopy equivalence is a homotopy equiva­ lence. Examples in the sequel will show that the converses are not always true. Hence, for any given family F of topological spaces, every topology type of F is contained in some isotopy type of F, and every isotopy type of F is con­ tained in some homotopy type of F. Consequently, the topological classifica­ tion problem can break into three steps as follows: Step 1. Homotopy classification. Determine effectively all of the homotopy types of the family F. Step 2. Isotopy classification. For each homotopy type a of the family F, determine effectively all of the isotopy types of the family a. Step 3. Topological classification. For each isotopy type /3 of the family F, determine effectively all of the topology types of the family (3 and exhibit a representative space in each of the topology types. In order to carry out the three steps of the topological classification problem for a given family F of topological spaces, one must make use of the various properties of spaces which are preserved by homotopy equivalences, isotopy equivalences, and homeomorphisms respectively. These properties are called the homotopy properties, the isotopy properties, and the topological properties respectively. It follows that every homotopy property is an isotopy property and that every isotopy property is a topological property. Examples in the sequel will show that the converses of these implications do not always hold. The main purpose of the present paper is to give general tests for homotopy and isotopy properties in terms of hereditary and weakly hereditary properties with the elementary properties in general topology as illustrations. These will Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 00:23:12, subject to the Cambridge Core terms of use. HOMOTOPY AND ISOTOPY PROPERTIES 169 be given in §§ 2 and 3. In the final section of the paper, we will describe a general method of constructing new homotopy and isotopy properties out of old ones as a striking and profound synthesis of various isolated known results. 2. Homotopy properties. A property P of topological spaces is called a homotopy property provided that it is preserved by all homotopy equivalences. Precisely, P is a homotopy property if and only if, for an arbitrary homotopy equivalence/: X —> F, that X has P implies that Y also has P. If a homotopy property P is given in the form of a number, a set, a group, or some other similar object, P is said to be a homotopy invariant. Some of the elementary properties in general topology are homotopy proper­ ties. As examples, one can easily prove the following assertions. PROPOSITION 2.1. Contractibility is a homotopy property of topological spaces. PROPOSITION 2.2. The cardinal number of components of a topological space X is a homotopy invariant. COROLLARY 2.3. Connectedness is a homotopy property of topological spaces. PROPOSITION 2.4. The cardinal number of path-components of a topological space X is a homotopy invariant. COROLLARY 2.5. Pathwise connectedness is a homotopy property of topological spaces. Nevertheless, most properties studied in general topology are not homotopy properties. To demonstrate this fact, let us first introduce the notion of weakly hereditary properties. A property P of topological spaces is said to be hereditary if each subspace of a topological space with P also has P; it is said to be weakly hereditary if every closed subspace of a topological space with P also has P. For examples, the following properties of a topological space X are weakly hereditary: (A) X is a TYspace, that is, every point in X forms a closed set of X. (B) X is a Hausdorff space. (C) X is a regular space. (D) X is a completely regular space. (E) X is a discrete space, that is, every set in X is open. (F) X is an indiscrete space, that is, the only open sets in X are the empty set • and the set X itself. (G) X is a metrizable space. (H) The first axiom of countability is satisfied in X, that is, the neighbour­ hoods of any point in X have a countable basis. (I) The second axiom of countability is satisfied in X, that is, the open sets of X have a countable basis. Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 00:23:12, subject to the Cambridge Core terms of use. 170 SZE-TSEN HU (J) X can be imbedded in a given topological space Y. (K) For a given integer n > 0, dim X < n. Here, the inductive dimension dim X is defined as follows: dim X = — 1 if X is empty, and dim X < n if for every point p (z X and every open neighbourhood U oî p there exists an open neighbourhood F C U of £ such that dim dV K n — 1, where dV denotes the boundary 7\7of V in X (2, p. 153). (L) X is a normal space. (M) X is a compact space. (N) X is a Lindelof space, that is, every open covering of X has a countable subcovering. (O) X is a paracompact space. (P) X is a locally compact space. (Q) F°r a given integer n > 0, Dim X < ?z. Here, the covering dimension Dim X is defined as follows: Dim X < n if every finite open covering of X has a refinement of order < n (2, p. 153). The first eleven properties (A)-(K) listed above are also hereditary. A topological space X is said to be a singleton space if X consists of a single point.
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